I have the differential equation $$ \epsilon y'' - x^2 y' - y = 0 $$ and I want to test if there is a boundary layer at $x = 1$ for $\epsilon \ll 1$.
I started with the rescaling $$ u = \dfrac{x - 1}{\delta}\ ,\qquad \delta = \delta(\epsilon) $$ which gives me the differential equation $$ \dfrac{\epsilon}{\delta^2} Y'' - \dfrac{1}{\delta}Y' - 2uY' - \delta u^2 Y' - Y = 0 $$ where $Y$ is now a function of $u$.
Now I'm supposed to apply a dominant balance argument to get $\delta(\epsilon)$, but I'm not sure how to do it. I'm used to apply dominant balance with three terms, such that I can guess two of them are of the same order and much greater than the third one. This usually leads to one consistent relation that gives the expression for $\delta(\epsilon)$. However, this equation presents 5 different terms, and I'm not sure how I should proceed to make such comparisons. Should I compare two against the others, four against one, ...
One method is to assume a power relation $\delta = \epsilon^\alpha$ and assume all other coefficients are order 1. This leads to five terms to play with $$ \epsilon^{1-2\alpha}, \epsilon^{-\alpha},1,\epsilon^\alpha,1. $$ As you mentioned, you have more freedom and what you choose is up to you. Follow through on each to see how it behaves.
Case 1: Balance the first two - This leads to $\alpha = 1$ and the first two terms dominate (some exponential decay).
Case 2: Balance first and third - This leads to $\alpha = 1/2$ and the second term dominates ($Y \sim const.$ in this scaling).
Case 3: Balance first and fourth - This leads to $\alpha = 1/3$ and similar behavior as case 2.
Are there any other cases?